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Variational derivation and compatible discretizations of the Maxwell-GLM system

Michael Dumbser, Alessia Lucca, Ilya Peshkov, Olindo Zanotti

Abstract

We present a novel variational derivation of the Maxwell-GLM system, which augments the original vacuum Maxwell equations via a generalized Lagrangian multiplier approach (GLM) by adding two supplementary acoustic subsystems and which was originally introduced by Munz et al. for purely numerical purposes in order to treat the divergence constraints of the magnetic and the electric field in the vacuum Maxwell equations within general-purpose and non-structure-preserving numerical schemes for hyperbolic PDE. Among the many mathematically interesting features of the model are: i) its symmetric hyperbolicity, ii) the extra conservation law for the total energy density and, most importantly, iii) the very peculiar combination of the basic differential operators, since both, curl-curl and div-grad combinations are mixed within this kind of system. A similar mixture of Maxwell-type and acoustic-type subsystems has recently been also forwarded by Buchman et al. in the context of a reformulation of the Einstein field equations of general relativity in terms of tetrads. This motivates our interest in this class of PDE, since the system is by itself very interesting from a mathematical point of view and can therefore serve as useful prototype system for the development of new structure-preserving numerical methods. Up to now, to the best of our knowledge, there exists neither a rigorous variational derivation of this class of hyperbolic PDE systems, nor do exactly energy-conserving and asymptotic-preserving schemes exist for them. The objectives of this paper are to derive the Maxwell-GLM system from an underlying variational principle, show its consistency with Hamiltonian mechanics and special relativity, extend it to the general nonlinear case and to develop new exactly energy-conserving and asymptotic-preserving finite volume schemes for its discretization.

Variational derivation and compatible discretizations of the Maxwell-GLM system

Abstract

We present a novel variational derivation of the Maxwell-GLM system, which augments the original vacuum Maxwell equations via a generalized Lagrangian multiplier approach (GLM) by adding two supplementary acoustic subsystems and which was originally introduced by Munz et al. for purely numerical purposes in order to treat the divergence constraints of the magnetic and the electric field in the vacuum Maxwell equations within general-purpose and non-structure-preserving numerical schemes for hyperbolic PDE. Among the many mathematically interesting features of the model are: i) its symmetric hyperbolicity, ii) the extra conservation law for the total energy density and, most importantly, iii) the very peculiar combination of the basic differential operators, since both, curl-curl and div-grad combinations are mixed within this kind of system. A similar mixture of Maxwell-type and acoustic-type subsystems has recently been also forwarded by Buchman et al. in the context of a reformulation of the Einstein field equations of general relativity in terms of tetrads. This motivates our interest in this class of PDE, since the system is by itself very interesting from a mathematical point of view and can therefore serve as useful prototype system for the development of new structure-preserving numerical methods. Up to now, to the best of our knowledge, there exists neither a rigorous variational derivation of this class of hyperbolic PDE systems, nor do exactly energy-conserving and asymptotic-preserving schemes exist for them. The objectives of this paper are to derive the Maxwell-GLM system from an underlying variational principle, show its consistency with Hamiltonian mechanics and special relativity, extend it to the general nonlinear case and to develop new exactly energy-conserving and asymptotic-preserving finite volume schemes for its discretization.

Paper Structure

This paper contains 22 sections, 3 theorems, 135 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. The augmented Maxwell-GLM system
  3. Governing equations and extra conservation law
  4. Symmetric hyperbolicity
  5. Formal asymptotic limit for $c_h \to \infty$
  6. Underlying variational principle
  7. Variational formulation for arbitrary nonlinear Lagrangian
  8. Hamiltonian nature of the Maxwell-GLM system
  9. Extension to special relativity
  10. Pure Maxwell
  11. The relativistic Maxwell-GLM
  12. First couple of Maxwell-GLM
  13. Second couple of Maxwell-GLM
  14. Compatible numerical discretizations
  15. Exactly energy-conserving semi-discrete numerical schemes on collocated meshes
  16. ...and 7 more sections

Key Result

Theorem 4.1

The finite volume scheme eqn.flux2dfv with a numerical flux $\mathbf{f}^{\ell\mathcal{r}}$ that satisfies eqn.compatibility_2dFV conserves total energy in the sense that, for vanishing boundary fluxes, we have

Figures (3)

  • Figure 1: Schematic representation of the primary $\Omega_c$ and of the dual $\Omega_p$ mesh.
  • Figure 2: Temporal evolution of the relative total energy error $\mathcal{E}^n/\mathcal{E}^0 - 1$ for a sequence of successively refined uniform grids composed of $N\times N$ elements obtained with the semi-discrete HTC scheme (left panel) as well as the fully-discrete semi-implicit scheme (right panel).
  • Figure 3: Left panel: Temporal evolution of the relative total energy error $\mathcal{E}^n/\mathcal{E}^0 - 1$ for initial data that is compatible with the vacuum Maxwell equations (black line), for general initial data employing the semi-implicit scheme (red line) as well as the HTC scheme (blue line) and for general initial data in case the nonlinear total energy potential \ref{['eqn.totalenergy_nonlin']} is used and the problem solved with the HTC scheme (green line). Right panel: time series of the divergence errors of the magnetic and electric field for the SIMM scheme and for initial data that is compatible with the vacuum Maxwell equations.

Theorems & Definitions (6)

  • Theorem 4.1: Total energy conservation
  • proof
  • Theorem 4.2
  • proof
  • Theorem 4.3
  • proof